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Model reduction techniques have been widely used to study the collective behavior of globally coupled oscillators. However, most approaches assume that there are infinitely many oscillators. Here we propose a new ansatz, based on the collective coordinate approach, that reproduces the collective dynamics of the Kuramoto model for finite networks to high accuracy, yields the same bifurcation structure in the thermodynamic limit of infinitely many oscillators as previous approaches, and additionally captures the dynamics of the order parameter in the thermodynamic limit, including critical slowing down that results from a cascade of saddle-node bifurcations.
The behavior at bifurcation from global synchronization to partial synchronization in finite networks of coupled oscillators is a complex phenomenon, involving the intricate dynamics of one or more oscillators with the remaining synchronized oscillat
The Kuramoto-Sakaguchi model for coupled phase oscillators with phase-frustration is often studied in the thermodynamic limit of infinitely many oscillators. Here we extend a model reduction method based on collective coordinates to capture the colle
We explore chaos in the Kuramoto model with multimodal distributions of the natural frequencies of oscillators and provide a comprehensive description under what conditions chaos occurs. For a natural frequency distribution with $M$ peaks it is typic
We analyze accuracy of different low-dimensional reductions of the collective dynamics in large populations of coupled phase oscillators with intrinsic noise. Three approximations are considered: (i) the Ott-Antonsen ansatz, (ii) the Gaussian ansatz,
Different types of synchronization states are found when non-linear chemical oscillators are embedded into an active medium that interconnects the oscillators but also contributes to the system dynamics. Using different theoretical tools, we approach